If [tex]f(x)=2 x-6[/tex] and [tex]g(x)=3 x+9[/tex], find [tex](f+g)(x)[/tex]
A. [tex](f+g)(x)=5 x+15[/tex]
B. [tex](f+g)(x)=x+15[/tex]
C. [tex](f+g)(x)=5 x+3[/tex]
D. [tex](f+g)(x)=-x-15[/tex]
Answer (2)
Add the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) .
Substitute the given expressions: ( f + g ) ( x ) = ( 2 x − 6 ) + ( 3 x + 9 ) .
Combine like terms: ( f + g ) ( x ) = ( 2 x + 3 x ) + ( − 6 + 9 ) .
Simplify the expression: ( f + g ) ( x ) = 5 x + 3 , so the final answer is 5 x + 3 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = 2 x − 6 and g ( x ) = 3 x + 9 , and we want to find ( f + g ) ( x ) . The sum of two functions is defined as ( f + g ) ( x ) = f ( x ) + g ( x ) .
Adding the Functions To find ( f + g ) ( x ) , we add the expressions for f ( x ) and g ( x ) : ( f + g ) ( x ) = f ( x ) + g ( x ) = ( 2 x − 6 ) + ( 3 x + 9 )
Combining Like Terms Now, we combine like terms: ( f + g ) ( x ) = ( 2 x + 3 x ) + ( − 6 + 9 )
Simplifying the Expression Finally, we simplify the expression: ( f + g ) ( x ) = 5 x + 3
Final Answer Therefore, ( f + g ) ( x ) = 5 x + 3 . Comparing this with the given options, we see that the correct answer is C.
Examples
Understanding how to combine functions is useful in many real-world scenarios. For example, if you have a business where your revenue R ( x ) depends on the number of items sold x , and your cost C ( x ) also depends on x , then the profit P ( x ) can be found by subtracting the cost from the revenue: P ( x ) = R ( x ) − C ( x ) . This is an example of combining functions to model a real-world situation. Similarly, if you have two different investment accounts, f ( t ) and g ( t ) , where t is time, the total value of your investments is ( f + g ) ( t ) .
To find ( f + g ) ( x ) , we add the functions: f ( x ) = 2 x − 6 and g ( x ) = 3 x + 9 . The result is ( f + g ) ( x ) = 5 x + 3 , which matches option C. Hence, the final answer is option C: ( f + g ) ( x ) = 5 x + 3 .
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